Ann Oper ResDOI 10.1007/s10479-016-2297-y

RISK MANAGEMENT APPROACHES IN ENGINEERING APPLICATIONS

Market-reaction-adjusted optimal central bankintervention policy in a forex market with jumps

Sandun Perera1 · Winston Buckley2 · Hongwei Long3

© Springer Science+Business Media New York 2016

Abstract Impulse controlwith randomreaction periods (ICRRP) is used to derive a country’soptimal foreign exchange (forex) rate intervention policy when the forex market reacts tothe interventions. This paper extends the previous work on ICRRP by incorporating a multi-dimensional jump diffusion process to model the state dynamics, and hence, enhance theviability of the extant model for applications. Furthermore, we employ a novel minimum costoperator that simplifies the computations of the optimal solutions. Finally, we demonstratethe efficacy of our framework by finding a market-reaction-adjusted optimal central bankintervention (CBI) policy for a country. Our numerical results suggests that market reactionsand the jumps in the forex market are complements when the reactions increase the forexrate volatility; otherwise, they are substitutes.

Keywords Optimal central-bank/government intervention policy · Financial marketreactions · Jump diffusions · Stochastic control

1 Introduction

Foreign exchange (forex) market interventions initiated by a central bank of a country (e.g.,Federal Reserve Bank of New York in U.S.) trigger reactions in the market, and thus, affectthe dynamics of the forex rate process for a random amount of time; this fact is also stronglysupported by empirical evidence.1 Bensoussan et al. (2012) study this phenomenon and

1 Empirical evidence on market reactions to central bank interventions can be found in Beine et al. (2002),Beine et al. (2003), Bonser-Neal and Tanner (1996), Caporale and Doroodian (2001), Dominguez (1998),Hung (1997), Mundaca (2001), Mundaca (2011) and Wilfling (2009).

B Sandun Pererasperera@umich.edu

1 School of Management, University of Michigan–Flint, Flint, MI 48502, USA

2 Department of Mathematical Sciences, Bentley University, Waltham, MA 02452, USA

3 Department of Mathematical Sciences, Florida Atlantic University, Boca Raton, FL 33431, USA

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theoretically incorporate the market reactions into the existing impulse control models—Cadenillas and Zapatero (1999), Mundaca and Øksendal (1998)—which are established tofind an optimal central bank intervention (CBI) policy for a country. A continuous diffusionprocess is used as the underlying stochastic process in Bensoussan et al. (2012), and therefore,the exchange rate is modeled by a geometric Brownian motion in the CBI model. However,the extant finance literature has accepted jump-diffusion as the primary driver of forex ratedynamics for over 25 years. Bates (1988) and Jorion (1988)were among the first to assert this;a literature review with more references on the significance of jumps in modeling exchangerate processes is presented at the end of this section. According to the literature, ignoringjumps in the forex rate processes can lead to serious consequences. Thus, the optimizationproblem of finding the best CBI policy under market reactions, hereafter referred to as theCBI problem with market reactions, should be modeled using a jump diffusion process inorder to capture the true dynamics of the exchange rate. To facilitate this, we extend the extantimpulse control model with random reaction periods by incorporating a multi-dimensionaljump diffusion process. This is the main contribution of our work.

Furthermore, the verification theorem presented in this paper generalizes the standardverification result in Øksendal and Sulem (2007) for the jump diffusion case. In particular,the verification theorem in Øksendal and Sulem (2007) does not cover models where thecontroller’s actions affect the dynamics of the state process for a random amount of timewhereas our verification theorem does embrace suchmodels. Ourmodel also employs a novelminimum cost operator that simplifies the computations of the optimal solutions, especiallyin the jump diffusion case.We demonstrate this through an example in Sect. 3. Moreover, this(probabilistic) minimum cost operator allows us to use a completely probabilistic approachin the proof of the verification theorem in contrast to the analytical approach suggestedin Bensoussan et al. (2012); this simplifies the elaborate proof of the theorem to a greatextent.

The rest of the paper is organized as follows: A literature review of forex rate modelswith jump-diffusion processes is presented next. In Sects. 2.1–2.2, we introduce our model,Quasi Integro-Variational Inequalities (QIVI) and QIVI-control. Section 2.3 presents theverification theorem for Impulse Control with RandomReaction Periods (ICRRP).WemodelaCBI problemwithmarket reactionswhen the exchange rate follows a jump diffusion processin Sect. 3.1; the solution of the corresponding QIVI is presented in Sect. 3.2. In Sect. 3.3, weemploy the Kou jump diffusion model (Kou 2002) to illustrate the computational procedureof the optimal solutions for the CBI problem, and then use the solutions to analyze the impactof jumps (in the forex rate process) and market reactions on the optimal CBI policy. Section4 concludes the paper, and provides possible future applications of our model. The proof ofthe verification theorem is given in the Appendix.

Foreign exchange rate as a jump-diffusion process: a literature review

Jump-diffusion processes have been applied in finance to capture discontinuous behaviorin asset prices because jump risks cannot be ignored in the pricing of financial assets. Sta-tistically, jump-diffusion nests diffusion as a special case. Being a more general model,jump-diffusion gives a better approximation of the data generating process as measure by,for example, the Kullback-Leibler information criterion (White 1982). Merton (1976) pio-neered the use of jump processes in continuous-time finance by deriving an option pricingwhen the underlying stock returns are generated by a mixture of both continuous and Poisson

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jump processes. Jump diffusion processes have been used to model exchange rates for manyyears; beginning with Bates (1988) and Jorion (1988), the finance literature has acceptedjump diffusions as the primary driver of forex rate dynamics.

Jorion (1988) considers a tractable model that combines both ARCH and jump processesfor foreign exchange dynamic. His results show that exchange rates have systematic discon-tinuities even after allowing for conditional heteroscedasticity in the diffusion process, andthat ignoring jumps can lead to serious mispricing errors for currency options. In anotherempirical study, Akgiray and Booth (1988) show that the mixed diffusion-jump process issuperior to the stable laws or a mixture of normal distributions as a model of exchange ratechanges for the British pound, French franc, and theWest Germanmark relative to the UnitedStates dollar. Merton’s jump-diffusion model is used in Svensson (1992), to study the foreignexchange risk premium in an exchange rate target zone regime with devaluation/realignmentrisks. Ahn and Thompson (1992) develop a theoretical jump-diffusion model for exchangerateswhich result from jumps in inflation in the domestic economies; theirmodel is consistentwith the results reported by empirical studies in Park et al. (1993). Ball and Roma (1993) andNieuwland et al. (1994) study jumps in exchange rate returns in European Monetary System(EMS) markets. They show that jumps, time-varying parameters and conditional leptokur-tosis are pertinent features in the empirical distributions of EMS exchange rate returns. Theempirical investigation in Bates (1996) shows that the US Dollar/Deutschemark exchangerate followsGeometric JumpDiffusion, where the instantaneous conditional variance followsa mean reverting square root process. Jiang (1998) proposes the simulation-based indirectinference approach to the estimation of general parametric continuous-time jump-diffusionprocesses from discretely observed data; applications to currency exchange rate models areundertaken and the results suggest that jumps are important components of the currencyexchange rate dynamics even when conditional heteroscedasticity and mean-reversion aretaken into account.

Doffou and Hilliard (2001) extend Bates (1996) model by investigating the effects of sto-chastic interest rates and jumps in the spot exchange rate on the pricing of currency futures,forwards and futures options. A jump-diffusion model for exchange rates in a target zonemodel is studied in Jong et al. (2001). Ahn et al. (2007) present explicit formulas for Euro-pean foreign exchange put and call options when the exchange rate dynamics are governedby jump-diffusion processes, while Yu (2007) provides closed form likelihood approxima-tions for multivariate jump-diffusions processes that are then applied to the realignment riskof the Chinese Yuan [see also Dumas et al. (1995)]. Guo and Hung (2007) use quadraticapproximation to price American exchange rate options under stochastic volatility, stochas-tic interest rates, and double jumps; they find evidence that jumps can have a critical impacton early exercise premiums that will be significant for deep out-of-the-money options withshort maturities. Nirei and Sushko (2011) examine a mechanism that generates tail events inthe distribution of foreign exchange returns and find that jumps in daily return of JapaneseYen against US Dollar over the period from January 1, 1999 through February 1, 2007 aremore likely to follow an exponential dampened power-law than Merton’s compound Pois-son process. In another study, Buncak (2013) reviews the role of Lévy jump processes inexchange rate modeling, which includes standard Brownian motion, Merton and Kou jump-diffusions; he, like many other researchers, confirms that continuous models are inferior intheir modeling of exchange rates. More recently, Chiang et al. (2016) provide a theoreticalexploration of currency options pricing under the presence of interest-rate regime shifts andexchange-rate asymmetric jumps driven by a double exponential jump diffusion process.

Finally, it should be mentioned that the jump-diffusion models allow for only a finitenumber of jumps in a finite time interval. However, there are other type of financial models

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which allow the driving Lévy processes to be of infinite activity, that is, having infinitelymany jumps in any finite time interval. Some frequently used models in finance are theNormal Inverse Gaussian (NIG) model of Barndorff-Nielsen (1997), Variance Gamma (VG)model of Madan and Seneta (1990) and Madan et al. (1998), Meixner model of Schoutensand Teugels (1998), CGMY model of Carr et al. (2002), and the finite moment log stablemodel of Carr and Wu (2003). We refer to Schoutens (2003) and Cont and Tankov (2004)for more detailed discussion of models based on Lévy processes. Recently, Aït-Sahalia andJacod (2011) propose statistical tests to discriminate between the finite and infinite activityof jumps in a semi-martingale discretely observed at high frequency; when implemented onhigh frequency stock returns, their tests point toward the presence of infinity activity jumpsin the data. Buncak (2013) discusses and compares multiple jump models such as Merton,Kou’s jump-diffusions, NIG, VG, and Meixner in the foreign exchange rates modeling. Hisempirical study shows that some of the infinite activity models such as NIG and Meixnermodels have slightly better goodness-of-fit than Kou’s jump-diffusions while VG model isworse than Kou’s model. It is quite feasible to extend the framework and results of our paperto stochastic models driven by Lévy processes with infinite activity although there will besome intractability issues.

2 A jump model for ICRRP

2.1 Model formulation

Let (�,F, P) be a complete probability space equipped with a filtration {Ft }t≥0 satisfyingthe usual assumptions. Consider a k-dimensional jump diffusion process X (t) of the form

d X (t) = μ(X (t))dt + σ(X (t))d B(t) +∫Rl

γ (X (t−), z)N (dt, dz), (1)

where X (0) = x ∈ Rk , μ : R

k → Rk , σ : R

k → Rk×m and γ : R

k × Rl → R

k×l

are measurable functions such that the solutions exist. Here B(t) is an m-dimensionalBrownian motion. The compensated Poisson random measure in the integral is definedas N (dt, dz)T = (N1(dt, dz1)), . . . , Nl(dt, dzl)), where N j (ds, dz j ) = N j (ds, dz j ) −ν j (dz j )ds for j = 1, . . . , l and {N j }l

j=1 are independent Poisson randommeasures with cor-

responding Lévymeasures ν j coming from l independent (1-dimensional) Lévyprocesses.2

2 Note that each column γ j of the k × l matrix γ = [γi j ] depends on z only through the j th coordinate z j ,i.e.

γ ( j)(t, z) = γ ( j)(t, z j ), j = 1, . . . , l; z = (z1, . . . , zl )T ∈ R

l .

Thus the integral on the right-hand side of (1) is just a shorthand matrix notation, i.e.,

∫Rl

γ (X (t−), z)N (dt, dz)

=⎛⎝ l∑

j=1

∫R

γ1 j (X (t−), z j )N j (dt, dz j ), . . . ,

l∑j=1

∫R

γk j (X (t−), z j )N j (dt, dz j )

⎞⎠

T

.

In component form (1) becomes:d Xd (t) = μd (X (t))dt +∑m

j=1 σd j (X (t))d B j (t) +∑lj=1

∫R

γd j (X (t−), z j )N j (dt, dz j ); 1 ≤ d ≤ k.

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If there are no interventions, the state of our system is a jump diffusion as in (1) with μ, σand γ satisfying the following two conditions for the existence and uniqueness of a solutionX (t).Condition 1 (At most linear growth): There exists a positive constant C1 < ∞ such that

‖σ(x)‖2 + |μ(x)|2 +∫R

l∑j=1

|γ ( j)(x, z j )|2ν j (dz j ) ≤ C1(1 + |x |2) for all x ∈ Rk .

Condition 2 (Lipschitz continuity): There exists a positive constant C2 < ∞ such that

‖σ(x) − σ(y)‖2 + |μ(x) − μ(y)|2 +l∑

j=1

∫R

|γ ( j)(x, z j ) − γ ( j)(y, z j )|2ν j (dz j )

≤ C2|x − y|2; for all x, y ∈ Rk .

Now assume that we are allowed to perform one of n different types of interventionsreferred to as type 1, type 2,…, type n, respectively. In particular, when the system is in statex at time t with the dynamics (1), and an impulse (i, ζ ) ∈ {1, 2, . . . , n} × Z i ⊂ N × R

p oftype i and size ζ is applied, where Z i is the set of admissible impulse values associated withthe intervention of type i , the state jumps immediately from X (t−) = x to X (t) = �i (x, ζ ),where �i : R

k × Z i → Rk is a given function. Moreover, the dynamics switches to

d Xi (t) = μi (Xi (t))dt + σ i (Xi (t))d B(t) +∫Rl

γ i (Xi (t−), z)N (dt, dz), (2)

where μi : Rk → R

k , σ i : Rk → R

k×m and γ i : Rk × R

l → Rk×l are given functions

satisfying at most linear growth and Lipschitz conditions for the existence and uniquenessof a solution Xi (t), and it persists for a random period of time T i . We denote the timeand the type of the j-th intervention by τ j and i j , respectively. For j = 1, 2, . . ., the j-th

intervention will affect the state dynamics until time τ j + Ti jj , where T

i jj ≥ 0 is a bounded

random variable for i j ∈ {1, 2, . . . , n}. The period(τ j , τ j + T

i jj

]is called the reaction

period resembling Bensoussan et al. (2012). The dynamics of the intervened process revertsto the pre-intervention dynamics at the end of the reaction period. We also assume thatT i

j ∼ T i for all j , where T i has the distribution function Fi , i = 1, 2, . . . , n, and T ij ’s are

iid and independent of both Bt and Nt , i.e., T ij is independent of the jump diffusion process.

Next, we provide the definition of an impulse control.

Definition 1 An impulse control is a double sequence u = (τ1, τ2, . . . , τ j , . . . ; (i1, ζ1),(i2, ζ2), . . . , (i j , ζ j ), . . .), where τ1 < τ2 < . . . are Ft -stopping times (the interventiontimes), and i1, i2, . . . and ζ1, ζ2, . . . are the corresponding intervention types and impulses,respectively, at these times such that each i j and ζ j are Fτ j -measurable.

We define the corresponding intervened state process X (u)x (t) with Xx (0) = x under an

impulse control u by

X (u)x (t) = Xx (t) ; 0 ≤ t < τ1, (3)

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X (u)x (τ j ) = �i j (X (u)

x (τ j−) + N Xx (τ j ), ζ j ); j = 1, 2, . . . , (4)

X (u)x (t) = X

i j

X (u)x (τ j )

(t) for τ j ≤ t ≤ τ j + Ti jj ; j = 1, 2, . . . , (5)

X (u)x (t) = X

X (u)x (τ j +T

i jj )

(t) for τ j + Ti jj < t < τ j+1; j = 1, 2, . . . , (6)

whereN X (t) is the jump of X resulting from the jump of the randommeasure N (t, ·) only.Note that we must distinguish between the (possible) jump of X (u)

x (τ j ) stemming from Nand the jump caused by the intervention u.

Now, let f : Rk → R be the running cost function. Moreover, suppose the cost of

making an intervention of type i with impulse ζ ∈ Z i is Ki (x, ζ ) when the state is x , whereKi : R

k × Z i → R is a given function. Then, we can define the performance criterion by

J (u)(x) = E x

⎡⎣∫ ∞

0e−r t f (X (u)(t))dt +

∞∑j=1

e−rτ j Ki j (X (u)(τ j−), ζ j )

⎤⎦ ,

where r > 0 is the discount rate and X (u)(τ j−) = X (u)(τ j−) + N X (τ j ). Here, E x

denotes the expectation operator with respect to the probability law of X (u) starting at x ,i.e., X (u)(0) = x . In order to minimize the expected total cost J (u) in a meaningful way, weintroduce the so-called admissible controls as follows.

Definition 2 The set U of impulse controls is called admissible, if for all x ∈ Rk , u ∈ U ,

(i) a unique solution X (u)x (t), t ≥ 0, of the system (3)–(6) exists,

(ii) limj→∞ τ j = ∞ a.s.,

(iii) E

[∫ ∞

0e−r t f (X (u)

x (t))dt

]< ∞,

(iv) E

⎡⎣ ∞∑

j=1

e−rτ j Ki j (X (u)x (τ j−), ζ j )

⎤⎦ < ∞, and

(v) τ j+1 − τ j ≥ Ti jj , j = 1, 2, . . ..

Observe that condition (v) in the above definition means that no interventions are possibleduring a reaction period and this assumption is required for tractability of our analysis.Although this assumption is imposed for tractability reasons, one can reasonably justifyits relevance for applications. For example, when we consider the CBI problem, it is clearthat the market reaction periods are designed to model relatively short-term re-adjustmentdurations, and therefore, these states are intended to be short-lived. After the central bankresets the exchange rate to the desired target level, there is a waiting period during which themedium term effects of the intervention is observed. If the bank believes that the market isstill in a transient reaction state (with uncertain parameters), but will soon revert to its formerdynamics (including volatility and drift), it is rational for the bank to preserve its capital bynot undertaking any further intervention until the exchange rate process has returned to itslong-term dynamical rate parameters.

We denote the value function of our impulse control problembyΦ. Namely, for all x ∈ Rk ,

Φ(x) = inf{

J (u)(x); u ∈ U}

= J (u∗)(x).

Our goal is to find the minimum cost Φ(x) and the corresponding optimal control u∗ ∈ U .

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2.2 Quasi integro-variational inequalities (QIVI)

We discuss the QIVI for impulse control of jump diffusions with random reaction periods inthis subsection. First, let us define a cost operator for an intervention of type i by

M(i, φ, x, ζ )= Ki (x, ζ )

+∫ ∞

0E�i (x,ζ )

[∫ T i

0e−rs f (Xi (s))ds+e−rT i

φ(Xi (T i ))

∣∣∣∣ T i = t

]d Fi (t),

where φ represents a generic function. Then, we can define the minimum cost operator Mi

for an intervention of type i and the minimum cost operator Mr for ICRRP as follows:

Miφ(x)= inf{

M(i, φ, x, ζ ); ζ∈Z i}

and Mrφ(x)=min{Miφ(x); i∈{1, 2, . . . , n}

}.

The cost operator Mr is equivalent to the cost operator in Bensoussan et al. (2012).However, the equivalent form thatwe propose in this papermakes computations of the optimalbands much easier. Moreover, as our CBI application in Sect. 3 illustrates, the computationsof the optimal bands are quite difficult when the underlying process follows a jump diffusionprocess. However, this (probabilistic) version of the operator makes those computationsrelatively simple. We now present the QIVI.

Definition 3 (QIVI) Let φ be a C1 function. Then, we say that φ satisfies the QIVI forICRRP, if for all x ∈ R

k , we have

φ(x) ≤ Mrφ(x),

Aφ(x) + f (x) ≥ 0,

[Aφ(x) + f (x)] [φ(x) − Mrφ(x)] = 0,

where3

Aφ(x) =k∑

i=1

μi (x)∂φ

∂xi(x) + 1

2

k∑i, j=1

(σσ T )i j (x)∂2φ

∂xi∂x j(x) − rφ(x)

+l∑

j=1

∫R

{φ(x + γ ( j)(x, z j )) − φ(x) − ∇φ(x) · γ ( j)(x, z j )}ν j (dz j ).

Davis et al. (2010) report the regularity properties of the value function for an infinite-horizon discounted cost impulse control problemwhen the underlying controlled process is amultidimensional jump diffusion. In particular, they show that if there is a sufficiently regularsolution of the above QIVI, then the solution is the value function. We refer the reader toDavis et al. (2010) for details.

Now, observe that a solution of the QIVI separates Rk into two disjoint regions. These

regions are defined by

C = {x ∈ Rk;φ(x) < Mrφ(x) and Aφ(x) + f (x) = 0}, and

I = {x ∈ Rk;φ(x) = Mrφ(x) and Aφ(x) + f (x) > 0},

3 The gradient operator ∇φ(x) denotes(

∂φ∂x1

,∂φ∂x2

, . . . ,∂φ∂xk

), and the dot product is defined by x · y =∑k

i=1 xi yi for x = (x1, x2, . . . , xk ) and y = (y1, y2, . . . , yk ).

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and they are called the continuation region and intervention region, respectively. Bensoussanet al. (2012) used a (continuous) diffusion process as their underlying process; hence, theprocess can not jump outside the continuation region. In contrast, we use a jump diffusionprocess as the underlying process in our model; this framework allows the process to jumpoutside the continuation region. We next construct the impulse control associated with theabove QIVI as follows:

Definition 4 (QIVI-control) Let φ be a solution of the QIVI. The QIVI-control u =(τ1, τ2, . . . , τ j , . . . ; (i1, ζ1), (i2, ζ2), . . . , (i j , ζ j ), . . .) (when it exists) is inductivelydefined by

τ0 = 0,

τ j+1 = inf{t > τ j + Ti jj ; X

(u j )x (t−) /∈ C}, and

i j+1 = argmin{Miφ(X

(u j )x (τ j+1−)); i ∈ {1, 2, . . . , n}

},

ζ j+1 = arg inf{

M(i j+1, φ, X(u j )x (τ j+1−), ζ ); ζ ∈ Z i j+1

},

where X(u j )x is the result of applying u j = (τ1, τ2, . . . , τ j ; (i1, ζ1), (i2, ζ2), . . . , (i j , ζ j )) to

Xx .

2.3 Verification theorem for ICRRP

A verification theorem for our problem is presented here. This theorem provides sufficientconditions for the existence of an optimal impulse control u∗ ∈ U and minimum cost Φ.

Theorem 1 Let φ be a solution of the QIVI. Suppose that for every x ∈ Rk , u ∈ U , we have

(i) E x [∫ ∞

01∂C

(X (u)(t)

)dt] = 0, where ∂C is the boundary of the set C,

(ii) ∂C is a Lipschitz surface (i.e., ∂C is locally the graph of a Lipschitz continuous func-tion),

(iii) φ ∈ C2(Rk \ ∂C) (i.e., φ(·) is a continuous real-valued function on Rk \ ∂C with

continuous second order derivatives) and the second order derivatives of φ are locallybounded near ∂C,

(iv) limt→∞ E x[e−r tφ

(X (u)(t)

)] = 0, and

(v) {φ−(X (u)x (τ )); τ is a stopping time} is uniformly integrable, then φ(x) ≤ Φ(x) for

all x ∈ Rk .

Suppose that we further have(vi) QIVI-control u corresponding to φ is admissible,

(vii) {φ(X (u)x (τ )); τ is a stopping time} is uniformly integrable, and

(viii) ζ = arg inf{M(i, φ, x, ζ ); ζ ∈ Z i } is a Borel measurable selection for each i , thenφ(x) = Φ(x) and u = u∗ for all x ∈ R

k .

The above verification theorem covers control problems in which the controller’s actionsaffect the state as well as the dynamics of the state process for a random amount of time;thus, the above result generalizes the verification result in Øksendal and Sulem (2007). Theproof of this theorem is presented in the Appendix.

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3 Application to central bank intervention (CBI)

In this section, we illustrate the procedure of finding the solution of the QIVI using a gen-eralization to a jump-diffusion of the CBI problem studied in Bensoussan et al. (2012). Thereader is referred to Bensoussan et al. (2012), Cadenillas and Zapatero (1999) and Mundacaand Øksendal (1998) for a literature review of the CBI problem.

3.1 CBI problem with market reactions in a Lévymarket

In the CBI problemwithmarket reactions, the central bank of a country is interested in findingthe optimal (i.e., least expensive) foreign exchange market intervention policy in order tokeep the country’s foreign exchange rate4 within a prescribed band or close to a target rateρ. There is a running cost for deviating away from this target rate, and there are both fixedand proportional costs associated with interventions.

Assume that Xx (t), the exchange rate at time t when there are no interventions, followsa jump diffusion of the form

d Xx (t)

Xx (t−)= μt + σ B(t) +

∫ t

0

∫R

η(z)N (ds, dz) ; Xx (0) = x, (7)

where x ∈ R and μ ∈ R, σ > 0 are constants, and η(z) is a function satisfying η(z) > −1.Some typical choices of η(·) are η(z) = ez − 1 with z ∈ R, and η(z) = θ z with z > −1/θfor θ > 0.

The central bank can perform one of two possible interventions; either to devalue thedomestic currency (type 1), or to increase the value of the domestic currency (type 2). Let�1(x, ζ ) = x + ζ , �2(x, ζ ) = x − ζ andZ1 = Z2 = [0,∞). If the bank’s j-th intervention,where j = 1, 2, · · · , is performed at time τ j is of type i ∈ {1, 2}, then the exchange ratedynamics immediately switches to

d Xi (t)

Xi (t−)= μi dt + σi d B(t) +

∫R

ηi (z)N (dt, dz), (8)

and persists until τ j + T ij , where T i

j ≥ 0 is a bounded random variable. The process reverts

to the pre-intervention dynamics when the market reactions end at τ j + T ij . Also observe that

Xi (τ j ) equals the latest pre-intervention exchange rate (including the possible jumps of theprocess itself) plus ζ (minus ζ ) for type 1 (type 2) intervention, respectively. The intervenedprocess with an impulse control u in the geometric Lévymarkets (7) and (8) is denoted byX (u)

x as in the previous section.The central bank’s objective is to find an admissible impulse control that minimizes the

functional J (u) defined by

J (u)(x) = E

⎡⎣∫ ∞

0e−r t f (X (u)

x (t))dt +∞∑j=1

e−rτ j Ki j (ζ j )

⎤⎦ ,

4 The number of domestic currency units per unit of foreign currency or a given basket of foreign currencies.

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where r is the discount rate, f (x) := (x − ρ)2 is the running cost, K1(ζ ) := C + cζ andK2(ζ ) := D +dζ are the corresponding costs of making an intervention of type 1 and type 2,respectively. Namely, we want to find an optimal control u∗ ∈ U which yields the minimumcost

Φ(x) = inf{

J (u)(x); u ∈ U}

= J (u∗)(x), for all x ∈ R.

3.2 Solution of the QIVI

We now solve the QIVI given in Sect. 2 for the CBI problem with market reactions. Notethat, in this case, the infinitesimal generator A and the minimum cost operator Mr can besimplified to the following forms:

Aφ(x) = μx∂φ

∂x+ 1

2σ 2x2

∂2φ

∂x2+∫R

[φ(x+η(z)x)−φ(x)−η(z)xφ

′(x)]ν(dz) − rφ(x),

Mrφ(x) = min{M1φ(x),M2φ(x)}, where

M1φ(x) = inf {C + cζ + R1(x + ζ ); ζ > 0} ,

M2φ(x) = inf {D + dζ + R2(x − ζ ); ζ > 0} ,

and Ri (x) = ∫∞0 E x

[∫ T i

0 e−rs f (Xi (s))ds + e−rT iφ(Xi (T i ))

∣∣∣∣ T i = t

]d Fi (t) for i ∈

{1, 2}.The solution to the QIVI for the CBI problem in Lévymarkets is similar to the solution to

QVI discussed in Bensoussan et al. (2012); thus, we will point out only the main differences.We still need to find a two-band control (see Fig. 1) characterized by four parameters L , l,w, and W such that 0 < L < l ≤ w < W < ∞.

Fig. 1 The optimal two-band policy

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Ann Oper Res

Under this policy, the optimal strategy is to stay in the band (L , W ), and bring the exchangerate down (respectively, up) to the levelw (respectively, l) whenever it reaches or jumps above(respectively, below) the level W (respectively, L). Thus, the cost function should satisfy

∀x ∈ (0, L] , Φ(x) = C + c(l − x) + R1(l), and

∀x ∈ [W,∞) , Φ(x) = D + d(x − w) + R2(w).

Moreover, the continuation and intervention regions respectively become C = (L , W ) andI = (0, L]∪ [W,∞) and sinceΦ is differentiable at the boundary of the continuation region,we have Φ ′(L) = −c and Φ ′(W ) = d . Furthermore, the function Φ(L) = C + c(z − L) +R1(z) (respectively, Φ(W ) = D + d(W − z) + R2(z)) is minimized at z = l (respectively,z = w); thus, the first order condition R′

1(l) = −c (respectively, R′2(w) = d) holds. Finally,

we assume thatAΦ(x) + f (x) = 0, ∀x ∈ (L , W ) . (9)

Similar to Bensoussan et al. (2012), we need to find Φ(x) ∈ C1 such that

Φ(x) =⎧⎨⎩

C + c(l − x) + R1(l), if x ≤ L ,

Φc(x), if L < x < W,

D + d(x − w) + R2(w), if x ≥ W,

(10)

in order to determine the QIVI characteristics associated with our QIVI-control problem inLévymarkets. Here Φc(x) is a solution to the Eq. (9); it can be easily seen (using standardtechniques of Boyce and Prima (1997)) that

Φc(x) = axα1 + bxα2 +(

1

r − σ 2 − 2μ − ∫R

η2(z)ν(dz)

)x2 −

(2ρ

r − μ

)x + ρ2

r,

where a and b are constants to be determined, and α1 < 0 and α2 > 0 are the roots of theequation g(α) = 0 with

g(α) = −r + μα + 1

2σ 2α(α − 1) +

∫R

[(1 + η(z))α − 1 − αη(z)

]ν(dz).

Note that, since g(.) is continuous and g(0) = −r < 0, the fact that limα→±∞ g(α) = ∞guarantees the existence of a positive root and a negative root of the equation g(α) = 0.Observe further that we require r − σ 2 − 2μ − ∫

Rη2(z)ν(dz) > 0 in order to derive Φc(·).

We can determine the unknown parameters L , l, w, W, a, and b by solving the followingsystem of equations

Φc(L) = C + c(l − L) + R1(l), (11)

Φc(W ) = D + d(W − w) + R2(w), (12)

Φ ′c(L) = −c, (13)

Φ ′c(W ) = d, (14)

R′1(l) = −c, (15)

R′2(w) = d. (16)

Using the above parameters, we then derive a feedback (i(x), ζ (x)) such that

(i(x), ζ (x)) ={

(1, l − x), if 0 < x ≤ L ,

(2, w − x), if x ≥ W,

and aQIVI-control u, inductively as inDefinition 4. Thus, our verification theoremestablishesthat J (u)(x) = Φ(x) for all x ∈ R and therefore u∗ = u.

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Note that Proposition 1 in Bensoussan et al. (2012) provides the conditions under whicha solution of the two-band control problem solves the QVI for the CBI problem in thegeometric Brownian markets. We remark that this proposition is still true for the QIVI of ourCBI problem in the Lévymarkets by simply using the new expressions for the generator A,R1 and R2. We omit the details here.

3.3 Computational example and its implications

In this subsection, we illustrate the computational procedure for finding the optimal solutionsof the CBI problem with market reactions using the Kou (2002) model as the jump-diffusionprocess. In theKoumodel, the distribution of jump sizes is an asymmetric double-exponentialdistribution with Lévydensity of the form

ν(dz) = γ[

pγ+e−γ+z I{z>0} + (1 − p)γ−e−γ−|z| I{z<0}]

dz,

with γ+ > 0, γ− > 0 governing the decay of the tails for the distribution of positive andnegative jump sizes, respectively, and p ∈ [0, 1] representing the probability of an upwardjump.Hereγ is the intensity of thePoissonprocess that counts the jumpsof the jump-diffusionprocess.

Our goal is to get an explicit formulation for the system of Eqs. (11)–(16), and then solve itnumerically. For this, we need to calculate Ri (x) and R

′i (x), i = 1, 2, explicitly.Moreover, we

want to simplify the expression ofΦc(·). For convenience, we choose η(z) = ηi (z) = ez −1.Then, some basic calculations show that∫

R

η2(z)ν(dz) = 2γ p

(γ+ − 2)(γ+ − 1)+ 2γ (1 − p)

(γ− + 1)(γ− + 2)< ∞

provided γ+ > 2 and γ− > 0. Note that g(α) is well-defined under the condition 0 < α < γ+or −γ− < α < 0. Again, simple calculations yield

g(α) = −r + μα + 1

2σ 2α(α − 1) + α(α − 1)γ p

(γ+ − α)(γ+ − 1)+ α(α − 1)γ (1 − p)

(γ− + α)(γ− + 1).

Observe that, for a given set of parameters, the roots of g(α) = 0 can now be computed, andhence we can write an explicit expression for Φc(·).

By solving (8) with ηi (z) = ez −1, it can be easily shown that the geometric Lévy processunder the intervention of type i is given by

Xix (t) = x exp

{(μi − σ 2

i

2−∫R

(ez − 1)ν(dz)

)t + σi B(t) +

∫ t

0

∫R

zN (ds, dz)

}.

Now let bi := μi − 12σ

2i − ∫

R(ez − 1)ν(dz) + ∫|z|<1 zν(dz). Then, we have

Xix (t) = x exp

{bi t + σi B(t) +

∫ t

0

∫|z|<1

z N (ds, dz) +∫ t

0

∫|z|≥1

zN (ds, dz)

}.

Using the exponential moments of Lévy process, it follows that, for u ∈ R,

E[(Xi

x (t))u]

= xu exp

{t

[bi u + 1

2σ 2

i u2 +∫R

(euz − 1 − uz1{|z|<1})ν(dz)

]}.

Next, we calculate Ri (x) for i = 1, 2. For simplicity, let us assume that the random reactiontime period T i ∼ Exponential(2), i.e., Fi (t) = 1 − e−t/2 for t > 0. Let � := (r − σ 2 −

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Ann Oper Res

2μ − ∫R

η2(z)ν(dz))−1 and Ci (u) := −r + bi u + 12σ

2i u2 + ∫

R(euz − 1− uz1{|z|<1})ν(dz).

Then, we have

Ri (x) =∫ ∞

0

{∫ t

0e−rs E[(Xi

x (s) − ρ)2] ds

+ e−r t E

[a(Xi

x (t))α1 + b(Xi

x (t))α2 + �(Xix (t))2 −

(2ρ

r − μ

)Xi

x (t) + ρ2

r

]}d Fi (t)

=∫ ∞

0

[x2

eCi (2)t − 1

Ci (2)− 2ρx

eCi (1)t − 1

Ci (1)+ ρ2 1 − e−r t

r

+ axα1eCi (α1)t + bxα2eCi (α2)t + �x2eCi (2) −(

2ρ

r − μ

)xeCi (1)t + ρ2

re−r t

]1

2e− t

2 dt

= axα1

1 − 2Ci (α1)+ bxα2

1 − 2Ci (α2)+ (� + 2)x2

1 − 2Ci (2)− 2ρx

1 − 2Ci (1)

(2 + 1

r − μ

)+ ρ2

r.

It follows that

R′i (x) = aα1xα1−1

1 − 2Ci (α1)+ bα2xα2−1

1 − 2Ci (α2)+ 2(� + 2)x

1 − 2Ci (2)− 2ρ

1 − 2Ci (1)

(2 + 1

r − μ

).

To get explicit formulas for Ci (u) where i ∈ {1, 2}, we define h(u) = ∫R(euz − 1 −

uz1{|z|<1})ν(dz). Then,

h(u) =∫ ∞

0(euz − 1 − uz1{|z|<1})γ pγ+e−γ+zdz

+∫ 0

−∞(euz − 1 − uz1{|z|<1})γ (1 − p)γ−eγ−zdz

= −γ p + γ pγ+γ+ − u

− γ pu(1 − e−γ+ − γ+e−γ+)

γ+

− γ (1 − p) + γ (1 − p)γ−γ− + u

+ γ (1 − p)u(1 − e−γ− − γ−e−γ−)

γ−

= −γ + γ

[pγ+

γ+ − u+ (1 − p)γ−

γ− + u

]

+ γ u

[(1 − p)(1 − e−γ− − γ−e−γ−)

γ−− p(1 − e−γ+ − γ+e−γ+)

γ+

]. (17)

Hence Ci (u) = −r + bi u + 12σ

2i u2 + h(u) with h(u) is given by (17). Furthermore, we can

simplify the expression of bi as follows:

bi = μi − 1

2σ 2

i −∫R

(ez − 1)(z)ν(dz) +∫

|z|<1zν(dz)

= μi − 1

2σ 2

i −∫R

(ez − 1 − z1{|z|<1})ν(dz)

= μi − 1

2σ 2

i − h(1).

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Ann Oper Res

Combining the above arguments and calculations, we can provide an explicit formulationof the system of Eqs. (11)–(16) with six unknowns a, b, L , l, w, and W as follows:

aLα1 + bLα2 + �L2 −(

2ρ

r − μ

)L = C + c(l − L) + alα1

1 − 2C1(α1)

+ blα2

1 − 2C1(α2)+ (� + 2)l2

1 − 2C1(2)− 2ρl

1 − 2C1(1)

(2 + 1

r − μ

)(18)

aW α1 + bW α2 + �W 2 −(

2ρ

r − μ

)W = D + d(W − w) + awα1

1 − 2C2(α1)

+ bwα2

1 − 2C2(α2)+ (� + 2)w2

1 − 2C2(2)− 2ρw

1 − 2C2(1)

(2 + 1

r − μ

)(19)

aα1Lα1−1 + bα2Lα2−1 + 2�L −(

2ρ

r − μ

)= −c (20)

aα1W α1−1 + bα2W α2−1 + 2�W −(

2ρ

r − μ

)= d (21)

aα1lα1−1

1 − 2C1(α1)+ bα2lα2−1

1 − 2C1(α2)+ 2(� + 2)l

1 − 2C1(2)− 2ρ

1 − 2C1(1)

(2 + 1

r − μ

)= −c.

(22)

aα1wα1−1

1 − 2C2(α1)+ bα2w

α2−1

1 − 2C2(α2)+ 2(� + 2)w

1 − 2C2(2)− 2ρ

1 − 2C2(1)

(2 + 1

r − μ

)= d.

(23)

It is not possible to find a closed-form solution to the above system. We therefore solveit numerically using Newton’s method. Next, we provide some interesting insights derivedfrom our numerical computations.

Impact of jumps on the optimal CBI policy

We first investigate the impact of Poisson jumps in the exchange rate on the optimal CBIpolicy when there is no reaction in the market. We use the same model parameters used inCadenillas and Zapatero (1999) as the parameters of our base model with no jumps or marketreactions. Thus, we let r = 0.06, ρ = 1.4, μ = μ1 = μ2 = 0.1, σ = σ1 = σ2 = 0.3,C = 0.5, c = 0.2, D = 0.7, and d = 0.4. The base case is compared with scenarios wherejump intensities are 3, 5 and 7. In all of these scenarios, we set p = 0.3, γ+ = 30 andγ− = 20. A comparison of the optimal values of the two-band control limits and the costfunctions are provided in Table 1 and Fig. 2.

Table 1 Optimal values of thetwo-band control limits whenthere is no market reactions

L l w W

Base case (no jumps) 0.551 1.082 1.227 2.387

Jump intensity γ = 3 0.534 1.080 1.232 2.426

Jump intensity γ = 5 0.524 1.078 1.235 2.450

Jump intensity γ = 7 0.514 1.076 1.238 2.473

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Ann Oper Res

0 0.5 1 1.5 2 2.5 3 3.5 45

5.5

6

6.5

7

7.5

8

Forex Rate

Cos

t Fun

ctio

n

Cost function with no jumpsCost function with jump intensity γ=3Cost function with jump intensity γ=5Cost function with jump intensity γ=7

Fig. 2 Cost functions when there is no market reactions

From Table 1, we observe that the optimal bands are wider when jumps exist; this meansthat the central bank would intervene less frequently when the jumps in the forex rate processare incorporated in the CBI model. Furthermore, the width of the optimal bands increaseswith the jump intensity; hence, the higher the jump intensity, the lesser the frequency ofinterventions by the central bank. Observe further, from Fig. 2, that the cost of the optimalpolicy is higher when jumps exist and it increases with the jump intensity.

Impact of market reactions on the optimal CBI policy

Market reactions can either increase or decrease the forex rate volatility. Let us first considerthe case where market reactions increase the forex rate volatility. We use the following modelparameters: r = 0.06, ρ = 1.4, μ = μ1 = μ2 = 0.1, σ = 0.15, σ1 = σ2 = 0.25, C = 0.5,c = 0.2, D = 0.7, d = 0.4, p = 0.3, γ = 3, γ+ = 30 and γ− = 20. In this case, ourobservations are similar to that of Bensoussan et al. (2012). Specifically, the optimal bands arewider and the cost of the optimal policy is higher when the market reactions are incorporated(see Table 2; Fig. 3).

Now, we turn our attention to the case where market reactions decrease the forex ratevolatility. Here, we consider four cases—base, jumps only, market reactions only, jumps andmarket reactions—for the comparison. The base case is still the same, i.e., use the samemodelparameters used in Cadenillas and Zapatero (1999). For “jumps only” case, we include jumpsto the base case with p = 0.3, γ = 5, γ+ = 30 and γ− = 20. In the “market reactions only”

Table 2 Optimal values of thecontrol limits when marketreactions increase the forex ratevolatility

L l w W

Without market reactions 0.616 1.037 1.134 2.099

With market reactions 0.558 1.360 1.414 2.212

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Ann Oper Res

0 0.5 1 1.5 2 2.5 3 3.53.5

4

4.5

5

5.5

6

6.5

7

7.5

Forex Rate

Cos

t Fun

ctio

n

Without market reactionsWith market reactions

Fig. 3 Cost functions when market reactions increase the forex rate volatility

case,we changeσ1 andσ2 of the base case to 0.15. Finally, “jumpsonly” and “market reactionsonly” cases are combined to create the “jumps and market reactions” case. Comparing thebase case with the “market reactions only” case, and the “jumps only” case with the “jumpsand market reactions” case, we conclude that the optimal bands are narrower and the cost ofthe optimal policy is lower when the market reactions are incorporated (see Table 3; Fig. 4).This observation is again parallel to that of Bensoussan et al. (2012). However, as evidentfrom Fig. 4, our results further suggest that the impact of market reactions is reduced by thepresence of jumps in the market.

Finally, combining our results with that of Bensoussan et al. (2012), wemake an importantobservation about the complementarity and substitutability between market reactions andjumps. For this, note that jumps always increase the optimal cost and make the optimal bandswider. However, market reactions have the same influence, i.e., increases the optimal costand make the optimal bands wider, only when it increases the forex rate volatility; otherwise,market reaction does the opposite. Hence, market reactions and the jumps in the forex marketare complements when the reactions increase the forex rate volatility; otherwise, they aresubstitutes.

Table 3 Optimal values of the control limits when market reactions decrease the forex rate volatility

L l w W

Base case (no jumps or market reactions) 0.551 1.082 1.227 2.387

Market reactions only 0.591 1.015 1.092 2.352

Jumps only 0.524 1.078 1.235 2.450

Jumps and market reactions 0.542 1.002 1.084 2.439

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Ann Oper Res

0 0.5 1 1.5 2 2.5 3 3.5 45

5.5

6

6.5

7

7.5

8

Forex Rate

Cos

t Fun

ctio

n

Base case (no jumps or market reactions)Market reactions onlyJumps onlyJumps and market reactions

Fig. 4 Cost functions when market reactions decrease the forex rate volatility

4 Concluding remarks and future applications

In this paper, we extend the existing impulse control model with random reaction periodsby incorporating a multi-dimensional jump diffusion process as the underlying stochasticprocess, and thus expand the practical usefulness of the extant model for applications suchas the CBI problem with market reactions. The new (probabilistic) minimum cost operatorintroduced in our QIVI makes the computations of the optimal solutions relatively easyfor applications; furthermore, the probabilistic nature of this operator allowed us to use acompletely probabilistic approach in the proof of the verification theorem, which simplifiesthe proof. Our verification theorem extends the classical verification result in Øksendal andSulem (2007), and therefore augments the existing literature. Finally, since we are able tonumerically solve a CBI problem, we derive some useful insights about the impact of jumpsand market reactions on the optimal CBI policy.

Other potential applications of ICRRP framework could range from classical cash man-agement (balance) problems to recently studied government debt ceiling/control problemssee Cadenillas and Huamán-Aguilar (2015) and Huamán-Aguilar and Cadenillas (2015).For example, the stochastic cash balance problem with fixed and proportional transactioncosts was first studied by Neave (1970) as an inventory problem where the stochastic natureappears in the demand for cash. Constantinides (1976) initiates the application of stochasticimpulse control to this problem and Constantinides and Richard (1978) is the first to showthe existence of a simple optimal impulse policy of a control band type. In their model, thecash inventory dynamic is a generalized Brownian motion. Bar-Ilan et al. (2004) incorporatea jump diffusion process in their cash inventory model to handle the exogenous risk due tounexpected shocks caused by financial crises. They argue that cash inventories linked withforex rates (e.g., exchange reserves of central banks) are greatly exposed to exogenous risk,and hence, they should be modeled using jump diffusion processes. Nascimento and Powell(2010) study a mutual fund cash management problem. Mutual funds have two main typesof investors—individual and institutional. Whenever a mutual fund manager deposits (with-

123

Ann Oper Res

draws) a large amount of cash into (from) his cash inventory, the investors in the mutualfund observe the manager’s actions and react to them. These reactions change deposit andwithdrawal patterns of the investors, and hence, change the dynamics of the demand processof the mutual fund’s cash inventory for a period of time. These reactions from the investorscan be captured using ICRRP. However, the cash inventory of a mutual fund is dependent onlarger redemption requests by institutional investors and bulky inflows of money from newinstitutional investors as well as the relatively small deposits and withdrawals by individualinvestors. Hence, the cash inventory level of a mutual fund is better captured by a jumpdiffusion process. In addition, if the mutual fund is also exposed to exogenous risk (e.g.,Forex Mutual Funds), then the cash inventory level of the fund must be modeled using ajump diffusion process. It therefore follows that mutual fund cash management problemswith market reactions can be modeled using the ICRRP framework presented in this paper.

Acknowledgments The authors thank the guest editors Jun-ya Gotoh and Stan Uryasev for their contribution,and the anonymous reviewers for their valuable comments and suggestions. Moreover, the authors are gratefulto Alain Bensoussan for his thoughtful suggestions that resulted in the improvement of this paper. We alsowould like to thank participants of the session of OM-Finance interface at INFORMS Annual Meeting 2011in Charlotte, session of Financial Services Section (Best Student Research Paper Competition) at INFORMSAnnual Meeting 2012 in Phoenix, OM seminar of the Jindal School of Management at the University ofTexas Dallas, Brown Bag Lunch Seminar of the Swiss Finance Institute, and Advances of ComputationalEconomics & Finance Seminar of the Institute of Operations Research at the University of Zurich for theirhelpful comments.

Appendix: Proof of the verification theorem

Let φ be a solution of the Q I V I ; then, φ ∈ C1. However, using (i)–(iii) and the Approxi-mation Theorem of Øksendal and Sulem (2007), we can and will assume that φ ∈ C2. Now,let u = (τ1, τ2, . . . , τ j , . . . ; (i1, ζ1), (i2, ζ2), . . . , (i j , ζ j ), . . .) ∈ U with τ0 = 0, and define

φd

(X (u)

x (s))

= e−rsφ(

X (u)x (s)

)for s ≥ 0. Then, since T

i jj is independent of X (u)

x for j ≥ 1

and i j ∈ {1, 2, . . . , n}, we have

E x[φd

(X (u)(τ j + T

i jj )) ∣∣∣∣T i j

j = t

]= E x

[φd

(X (u)(τ j + t)

)], (24)

where t ≥ 0.Next, we derive an expression for the right-hand side of (24). For this, consider

g(s, X (u)x (s)) := e−rsφ

(X (u)

x (s))

= φd

(X (u)

x (s))and observe that g(s, y) is C1 w.r.t.

s ≥ 0 (i.e., for a given y ∈ Rk , g(·, y) is a continuous real-valued function on [0,∞) with

continuous first order derivatives) and C2 w.r.t. y ∈ Rk (i.e., for a given s ≥ 0, g(s, ·) is a

continuous real-valued function on Rk with continuous second order derivatives). Therefore,

applying the multi-dimensional Itô ’s formula (cf. Øksendal and Sulem (2007)) for g from τ j

to τ j + t , we obtain5

φd

(X (u)

x (τ j + t))

= φd

(X (u)

x (τ j ))

− r∫ τ j +t

τ j

φd

(X (u)

x (s))

ds

+∫ τ j +t

τ j

k∑p=1

∂φd

∂x p

(X (u)

x (s)) [

μi jp

(X (u)

x (s))

ds + σi jp

(X (u)

x (s))

d B(s)]

5 Note that, from τ j to τ j + t , the process X (u) follows Xi j .

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Ann Oper Res

+1

2

∫ τ j +t

τ j

k∑p,q=1

(σ i j (σ i j )T )pq

(X (u)

x (s)) ∂2φd

∂x p∂xq

(X (u)

x (s))

ds

+∫ τ j +t

τ j

l∑p=1

∫R

{φd

(X (u)

x (s−) + (γ i j )(p)(X (u)x (s−), z p)

)− φd

(X (u)

x (s−))

−k∑

q=1

(γ i j )(p)q

(X (u)

x (s−), z p

) ∂φd

∂xq

(X (u)

x (s−))}νp(dz p)ds

+∫ τ j +t

τ j

l∑p=1

∫R

{φd

(X (u)

x (s−) + (γ i j )(p)(X (u)x (s−), z p)

)

−φd

(X (u)

x (s−))}

Np(ds, dz p).

Rearranging the terms in the above equation then yields

φd

(X (u)

x (τ j + t))

= φd

(X (u)

x (τ j ))

+∫ τ j +t

τ j

k∑p=1

∂φd

∂x p

(X (u)

x (t))

σi jp

(X (u)

x (s))

d B(s)

+∫ τ j +t

τ j

Ai j φd

(X (u)

x (s))

ds

+∫ τ j +t

τ j

l∑p=1

∫R

{φd

(X (u)

x (s−) + (γ i j )(p)(X (u)x (s−), z p)

)

−φd

(X (u)

x (s−))}

Np(ds, dz p), (25)

where, for x ∈ Rk ,

Ai j φd(x) :=k∑

p=1

μi jp (x)

∂φd

∂x p(x) + 1

2

k∑p,q=1

(σ i j (σ i j )T )pq(x)∂2φd

∂x p∂xq(x) − rφd(x)

+l∑

p=1

∫R

{φd(x+(γ i j )(p)(x, z j ))−φd(x)−∇φd(x) · (γ i j )(p)(x, z p)}νp(dz p).

Thus, since

E x

⎡⎣∫ τ j +t

τ j

k∑p=1

∂φd

∂x p

(X (u)(t)

)σ

i jp

(X (u)(s)

)d B(s)

⎤⎦ = 0

and

E x

⎡⎣∫ τ j +t

τ j

l∑p=1

∫R

{φd(X (u)(s−) + (γ i j )(p)(X (u)(s), z p))

−φd

(X (u)(s−)

)}Np(ds, dz p)

⎤⎦ = 0,

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Ann Oper Res

taking the expectation of both sides of (25), we have

E x[φd

(X (u)(τ j + t)

)]= E x

[φd

(X (u)(τ j )

)]+E x

[∫ τ j +t

τ j

Ai j φd

(X (u)(s)

)ds

]. (26)

Now, note that, from τ j + t to τ j+1, the process X (u) follows the original process X .Therefore, using an argument similar to that leading up to (26), it can be easily shown that

E x[φd

(X (u)(τ j+1−)

)]= E x

[φd

(X (u)(τ j + t)

)]+ E x

[∫ τ j+1

τ j +tAφd

(X (u)(s)

)ds

],

(27)where X (u)

x (τ j+1−) = X (u)x (τ j+1−) + N Xx (τ j+1).

It follows from (26) and (27) that

E x[φd

(X (u)(τ j )

)− φd

(X (u)(τ j+1−)

)]= −E x

[∫ τ j +t

τ j

Ai j φd

(X (u)(s)

)ds

]

− E x

[∫ τ j+1

τ j +tAφd

(X (u)(s)

)ds

]. (28)

Next, observe that∫ ∞

0E x[φd

(X (u)(τ j )

)− φd

(X (u)(τ j+1−)

)]d Fi j (t)

= E x[φd

(X (u)(τ j )

)− φd

(X (u)(τ j+1−)

)] ∫ ∞

0d Fi j (t)

= E x[φd

(X (u)(τ j )

)− φd

(X (u)(τ j+1−)

)],

where the first equality is due the independence. Therefore, integrating (28) with respect tothe probability measure induced by T i j on [0,∞) first and then summing from j = 1 toj = m, we obtain

m∑j=1

E x[φd(X (u)(τ j )) − φd

(X (u)(τ j+1−)

)]

= −m∑

j=1

∫ ∞

0E x

[∫ τ j +t

τ j

Ai j φd

(X (u)(s)

)ds

]d Fi j (t)

−m∑

j=1

∫ ∞

0E x

[∫ τ j+1

τ j +tAφd

(X (u)(s)

)ds

]d Fi j (t). (29)

Moreover, using an argument similar to that leading up to (26) again, it can be easilyobserved that

E x[φd

(X (u)(τ1−)

)]= E x

[φd

(X (u)

x (0))]

+ E x[∫ τ1

0Aφd

(X (u)(s)

)ds

]

= φ (x) + E x[∫ τ1

0Aφd

(X (u)(s)

)ds

]. (30)

123

Ann Oper Res

Combining (29) and (30) yield

φ (x) +m∑

j=1

E x[φd(X (u)(τ j )) − φd

(X (u)(τ j+1−)

)]− E x

[φd

(X (u)(τ1−)

)]

= −m∑

j=1

∫ ∞

0E x

[∫ τ j +t

τ j

Ai j φd

(X (u)(s)

)ds

]d Fi j (t) − E x

[∫ τ1

0Aφd

(X (u)(s)

)ds

]

−m∑

j=1

∫ ∞

0E x

[∫ τ j+1

τ j +tAφd

(X (u)(s)

)ds

]d Fi j (t).

Equivalently, we have

φ(x) +m∑

j=1

E x[φd(X (u)(τ j )) − φd(X (u)(τ j−))

]− E x

[φd(X (u)(τm+1−))

]

=m∑

j=1

∫ ∞

0E x

[∫ τ j +t

τ j

{−Ai j φd

(X (u)(s)

)}ds

]d Fi j (t)

+E x[∫ τ1

0

{−Aφd

(X (u)(s)

)}ds

]

+m∑

j=1

∫ ∞

0E x

[∫ τ j+1

τ j +t

{−Aφd

(X (u)(s)

)}ds

]d Fi j (t). (31)

Now, since φ satisfies the QIVI, we have f ≥ −Aφ. Therefore, fd ≥ −Aφd wherefd(X (u)(s)) = e−rs f (X (u)(s)). Then, it follows from (31) that

φ(x) +m∑

j=1

E x[φd(X (u)(τ j )) − φd(X (u)(τ j−))

]− E x

[φd(X (u)(τm+1−))

]

≤m∑

j=1

∫ ∞

0E x

[∫ τ j +t

τ j

{−Ai j φd

(X (u)(s)

)}ds

]d Fi j (t) + E x

[∫ τ1

0fd

(X (u)(s)

)ds

]

+m∑

j=1

∫ ∞

0E x

[∫ τ j+1

τ j +tfd

(X (u)(s)

)ds

]d Fi j (t). (32)

Next, since T i j is independent of Xi jx for i j ∈ {1, 2, . . . , n}, by the definition of Mr , we

have

Mr φ(X (u)x (τ j −)) ≤ Ki j (X (u)

x (τ j −), ζ j )

+∫ ∞

0E�i j (X (u)

x (τ j −),ζ j )

[∫ t

0e−rs f

(Xi j (s)

)ds + e−r tφ

(Xi j (t)

)]d Fi j (t)

But X (u)x (τ j ) = �i j (X (u)

x (τ j−), ζ j ), therefore

Mrφ(X (u)x (τ j−))≤ Ki j (X (u)

x (τ j−), ζ j )+∫ ∞

0E X (u)

x (τ j )

[∫ t

0e−rs f

(Xi j (s)

)ds

]d Fi j (t)

+∫ ∞

0E X (u)

x (τ j )[e−r tφ

(Xi j (t)

)]d Fi j (t). (33)

123

Ann Oper Res

Now, using the strongMarkov property and the fact that X (u)x (s) = X

i j

X (u)x (τ j )

(s) for τ j ≤ s ≤τ j + t , we observe that

∫ ∞

0E X (u)

x (τ j )

[∫ t

0e−rs f

(Xi j (s)

)ds

]d Fi j (t)

=∫ ∞

0

{∫ t

0e−rs E X (u)

x (τ j )[

f(

Xi j (s))]

ds

}d Fi j (t)

=∫ ∞

0

{∫ t

0e−rs E x

[f(

X (u)(τ j + s)) ∣∣∣∣Fτ j

]ds

}d Fi j (t)

=∫ ∞

0E x[∫ t

0e−rs f

(X (u)(τ j + s)

)ds

∣∣∣∣Fτ j

]d Fi j (t)

=∫ ∞

0erτ j E x

[∫ τ j +t

τ j

e−rs f(

X (u)(s))

ds

∣∣∣∣Fτ j

]d Fi j (t)

and

∫ ∞

0E X (u)

x (τ j )[e−r tφ

(Xi j (t)

)]d Fi j (t)

=∫ ∞

0E

[e−r tφ

(X

i j

X (u)x (τ j )

(τ j + t)

) ∣∣∣∣Fτ j

]d Fi j (t)

=∫ ∞

0E x[

e−r tφ(

X (u)(τ j + t)) ∣∣∣∣Fτ j

]d Fi j (t).

Then, from (33), we have

Mrφ(X (u)x (τ j−)) ≤ Ki j (X (u)

x (τ j−), ζ j )

+∫ ∞

0erτ j E x

[∫ τ j +t

τ j

e−rs f(

X (u)(s))

ds

∣∣∣∣Fτ j

]d Fi j (t)

+∫ ∞

0E x[

e−r tφ(

X (u)(τ j + t)) ∣∣∣∣Fτ j

]d Fi j (t).

Multiplying the above equation by e−rτ j yields

Mrφd(X (u)x (τ j−)) ≤ e−rτ j Ki j (X (u)

x (τ j−), ζ j )

+∫ ∞

0E x

[∫ τ j +t

τ j

fd

(X (u)(s)

)ds

∣∣∣∣Fτ j

]d Fi j (t)

+∫ ∞

0E x[φd

(X (u)(τ j + t)

) ∣∣∣∣Fτ j

]d Fi j (t). (34)

Next, taking the conditional expectation of both sides of (25) first and then using an argumentsimilar to that leading up to (26) again, we obtain

123

Ann Oper Res

E x[φd

(X (u)(τ j + t)

) ∣∣∣∣Fτ j

]

= E x[φd

(X (u)

x (τ j )) ∣∣∣∣Fτ j

]+ E x

[∫ τ j +t

τ j

Ai j φd

(X (u)(s)

)ds

∣∣∣∣Fτ j

].

= φd

(X (u)

x (τ j ))

+ E x

[∫ τ j +t

τ j

Ai j φd

(X (u)(s)

)ds

∣∣∣∣Fτ j

], (35)

where the second equality is due to the Fτ j -measurability of φd

(X (u)

x (τ j )).

Integrating (35) with respect to the probability measure induced by T i j on [0,∞), we have

∫ ∞

0E x[φd

(X (u)(τ j + t)

) ∣∣∣∣Fτ j

]d Fi j (t)

= φd

(X (u)

x (τ j ))

+∫ ∞

0E x

[∫ τ j +t

τ j

Ai j φd

(X (u)(s)

)ds

∣∣∣∣Fτ j

]d Fi j (t). (36)

Substituting (36) into (34) and subtracting φd(X (u)x (τ j−)) from both sides then yield

Mr φd (X (u)x (τ j−)) − φd(X (u)

x (τ j−)) ≤ e−rτ j Ki j (X (u)x (τ j−), ζ j )

+∫ ∞

0Ex

[∫ τ j +t

τ j

fd

(X (u)(s)

)ds

∣∣∣∣Fτ j

]d Fi j (t)

+∫ ∞

0Ex

[∫ τ j +t

τ j

Ai j φd

(X (u)(s)

)ds

∣∣∣∣Fτ j

]d Fi j (t)

+ φd

(X (u)

x (τ j ))

− φd (X (u)x (τ j−)).

Now, taking the expectation of both sides of the above inequality first and then summing theresulting inequality from j = 1 to j = m, we obtain

m∑j=1

E x[Mrφd(X (u)

x (τ j−)) − φd(X (u)x (τ j−))

]

≤ E x

⎡⎣ m∑

j=1

e−rτ j Ki j (X (u)x (τ j−), ζ j )

⎤⎦

+m∑

j=1

∫ ∞

0E x

[∫ τ j +t

τ j

fd

(X (u)(t)

)dt

]d Fi j (t)

+m∑

j=1

∫ ∞

0E x

[∫ τ j +t

τ j

Ai j φd

(X (u)(t)

)dt

]d Fi j (t)

+m∑

j=1

E x[φd(X (u)(τ j )) − φd(X (u)(τ j−))

]. (37)

123

Ann Oper Res

Finally, it follows from (32) and (37) that

φ(x) +m∑

j=1

E x[Mrφd(X (u)(τ j−)) − φd(X (u)(τ j−))

]

≤ E x[φd(X (u)(τm+1−))

]−

m∑j=1

E x[φd(X (u)(τ j )) − φd(X (u)(τ j−))

]

−m∑

j=1

∫ ∞

0E x

[∫ τ j +t

τ j

Ai j φd

(X (u)(s)

)ds

]d Fi j (t) + E x

[∫ τ1

0fd

(X (u)(s)

)ds

]

+m∑

j=1

∫ ∞

0E x

[∫ τ j+1

τ j +tf(

X (u)(t))

dt

]d Fi j (t)

+E x

⎡⎣ m∑

j=1

e−rτ j Ki j (X (u)(τ j−), ζ j )

⎤⎦+

m∑j=1

∫ ∞

0E x

[∫ τ j +t

τ j

f(X (u)(s)

)ds

]d Fi j (t)

+m∑

j=1

∫ ∞

0E x

[∫ τ j +t

τ j

Ai j φd

(X (u)(s)

)ds

]d Fi j (t)

+m∑

j=1

E x[φd(X (u)(τ j )) − φd(X (u)(τ j−))

]

= E x[∫ τm+1

0fd(X (u)(s))ds

]+ E x

[φd(X (u)(τm+1−))

]

+E x

⎡⎣ m∑

j=1

e−rτ j Ki j (X (u)(τ j−), ζ j )

⎤⎦

= E x

⎡⎣∫ τm+1

0e−rs f (X (u)(s))ds + e−rτm+1φ(X (u)(τm+1−))

+m∑

j=1

e−rτ j Ki j (X (u)(τ j−), ζ j )

⎤⎦ . (38)

Since the second term on the left-hand side of (38) is non-negative, letting m → ∞ in(38) and using conditions (iv)-(v), we have

φ(x) ≤ E x

⎡⎣∫ ∞

0e−r t f (X (u)(t))dt +

∞∑j=1

e−rτ j Ki j (X (u)(τ j−), ζ j )

⎤⎦ = J (u)(x). (39)

Hence, we have that φ(x) ≤ Φ(x) = inf{J (u)(x); u ∈ U}.Moreover, if the QIVI-control u corresponding to φ is admissible, then we can apply the

above argument to u = (τ1, τ2, . . . , τ j , . . . ; (i1, ζ1), (i2, ζ2), . . . , (i j , ζ j ), . . .). Now, sinceAφ + f = 0 on the boundary of C, we obtain the equality in (32) and by the choices ofζi = ζi and i j = i j , we also have the equality in (34) and (37). Then, since condition (i)implies that the second term on the left-hand side of (38) goes to zero as m → ∞, we achieve

123

Ann Oper Res

the desired equality in (39). Therefore, we have φ(x) = J (u)(x). Hence, φ(x) = Φ(x) andu∗ = u. ��

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